
and \( \Vert \widehat{B}^{(k)} \widehat{\theta }_\text{ols}^* -\widehat{\Sigma }_{xy}^{(k)*} \Vert = \sum _{m=1}^{M} \Vert  \widehat{\Sigma }_{xx}^{(k, m)}\widehat{\theta }_\text{ols} - \widehat{\Sigma }_{xy}^{(k, m)} \Vert \) . For simplicity, we next assume that both the covariate \( X_i \)  and the random error \( \varepsilon _i \)  are normally distributed.
We are then able to study the asymptotic behavior of \( \max _{1\le k \le K} \Vert  \widehat{E}^{(k)}\Vert  \) .
To this end, we need to study \(\max _{1\le k\le K} \Vert P_1 \widehat{\Sigma }_{xx} ^{-1}P_1^\top \widehat{B}^{(k)} \Vert  \)  and \( \max _{1\le k\le K} \big \Vert \widehat{B}^{(k)} \widehat{\theta }_\text{ols}^* -\widehat{\Sigma }_{xy}^{(k)*} \Vert  \)  separately.
We first consider \(\max _{1\le k\le K} \Vert P_1 \widehat{\Sigma }_{xx} ^{-1}P_1^\top \widehat{B}^{(k)} \Vert  \) .
By Lemma A.3 in [1]}, there should exist constants \( C_1>0 \)  and \( C_2>0 \)  such that \( P(\max _{1\le m\le M}\Vert \widehat{\Sigma }_{xx}^{(k, m)} - \Sigma _{xx}\Vert _F \ge \nu )\le C_1\exp (-C_2 N\nu ^2) \)  for any sufficiently small \( \nu >0 \) , as long as \( M \)  and \( p \)  are fixed. Consequently, we should have \( P(\max _{m,k}\Vert \widehat{\Sigma }_{xx}^{(k, m)} - \Sigma _{xx}\Vert _F \ge \nu )\le C_1\exp (\log K-C_2 N\nu ^2) \) , which tends to 0 as long as \( \log K = o( N) \) .
Therefore, we can derive that \( \max _{m,k}\Vert \widehat{\Sigma }_{xx} ^{-1} \widehat{\Sigma }_{xx}^{(k, m)}\Vert  \le \Vert \widehat{\Sigma }_{xx} ^{-1}\Vert \max _{m,k}\Vert  \widehat{\Sigma }_{xx}^{(k, m)}\Vert _F =O_p(1) \) . As a consequence, we should have \( \max _{1\le k\le K} \Vert P_1 \widehat{\Sigma }_{xx} ^{-1}P_1^\top \widehat{B}^{(k)} \Vert =O_p(1) \)  as long as \( \log K = o(N) \) .
